Scheduling Problems with Financial Resource ...

Scheduling Problems with Financial Resource Constraints E. R. Gafarov∗, A. A. Lazarev∗, F. Werner† ∗ †

Institute of Control Sciences RAS, [email protected], [email protected] Otto-von-Guericke-Universit¨at Magdeburg, [email protected]

. . . < ty , of earnings of the resource. At each time ti , i = 0, 1, . . . , y, we receive an amount G(ti ) ≥ 0 of the resource. For each job j ∈ N , a consumption gj ≥ 0 of the resource arises when the job is

In a resource-constrained scheduling problem, one wishes to schedule the jobs in such a way that the given resource constraints are fulfilled and a given objective function attains its optimal value. In this paper, we deal with single machine scheduling problems with a non-renewable resource, such problems are also referred to as financial scheduling problems.

started. Thus, we have

n P j=1

gj =

y P

G(ti ).

i=0

Let Sj be the starting time of the processing of job j. A schedule S = (Sj1 , Sj2 , . . . , Sjn ) describes the order of processing the jobs: π = (j1 , j2 , . . . , jn ). Such an order is uniquely determined by a permutation (sequence) of the jobs of set N . A schedule S = (Sj1 , Sj2 , . . . , Sjn ) is feasible, if the machine processes no more than one job at a time and the resource constraints are fulfilled, i.e., for each i = 1, 2, . . . , n, we have

The research in the area of scheduling problems with a non-renewable resource is rather limited. In [1], some polynomially bounded algorithms are presented for scheduling problems with precedence constraints (not restricted to single machine problems). Some results for preemptive scheduling of independent jobs on unrelated parallel machines have been presented in [2]. Toker et al. [3] have shown that the problem of minimizing the makespan with a unit supply of a resource at each time period is polynomially solvable. Janiak et al. [4, 5] have considered single machine problems, in which the processing times or the release times depend on the consumption of a non-renewable resource.

i P k=1

gjk ≤

P

G(tl ).

∀l: tl ≤Sji

Moreover, we will call the sequence π a schedule too since one can compute S = (Sj1 , Sj2 , . . . , Sjn ) in O(n) time applying a list scheduling algorithm to the sequence π. Then Cjk (π) = Sjk + pjk denotes the completion time of job jk in schedule π. If Cj (π) > dj , then job j is tardy and we have Uj = 1, otherwise Uj = 0. Moreover, let Tj (π) = max{0, Cj (π)−dj } be the tardiness of job j in schedule π. We denote by Cmax = Cjn (π) the makespan of schedule π and by Lj (π) = Cj (π)−dj the lateness of job j in π.

The problems under consideration can be formulated as follows. We are given a set N = {1, 2, . . . , n} of n independent jobs that must be processed on a single machine. Preemptions of a job are not allowed. The machine can handle only one job at a time. All the jobs are assumed to be available for processing at time 0. For each job 1 Some Complexity Results j, j ∈ N , a processing time pj ≥ 0 and a due date dj are given. In addition, we have one non- Theorem 1 Problems 1|N R|Cmax , 1|N R, dj = P P renewable resource G (e.g. money, energy, etc.) d| Tj , 1|N R| Uj and 1|N R|Lmax are N P and a set of times {t0 , t1 , . . . , ty }, t0 = 0, t0 < t1 < hard in the strong sense, and the problem 1

1|N R|

P

j = 1, 2, . . . , n. Then both problems are equivalent. It can be noted that the special case 1|N R : P P αt = 1, pj = 0| Tj can be solved in O(n4 gj ) time by Lawler’s algorithm [8] since we obtain a P problem 1|| Tj with processing times gj . According to the definition in [7], we have LEN GT H[I] = n + y. In fact, the string x consists of 2n + 2y + 3 numbers. However, if we consider problem 1|N R : αt = 1, pj = P 0| Tj as a special case of problem 1|N R : P pj = p| Tj and use the same encoding scheme, P then LEN GT H[I] = n + y = n + gj , i.e., the length of the input is pseudo-polynomial. Since, as mentioned above, problem 1|N R : αt = P P 1, pj = 0| Tj can be solved in O(n gj ) time by Lawler’s algorithm, the complexity of this algorithm would polynomially depend on the inP put length LEN GT H[I] = n + gj . For this reason, we consider sub-problem 1|N R : αt = P 1, pj = p| Tj as a separate problem and use the encoding scheme e0 , in which we present an instance as a string ”p, d1 , d2 , . . . , dn , g1 , g2 . . . , gn ”, i.e., LEN GT H[I] = n. Let us now consider the sub-case of problem P 1|N R, pj = p| Tj , where the times of earnings of the resource are given by t1 = M, t2 = 2M, . . . , tn = nM and P G(ti ) = M for all i =

Cj is N P -hard in the ordinary sense. P

Theorem 2 The problem 1|N R, dj = d| Tj is not in AP X, where AP X is the class of optimization problems that allow polynomial-time approximation algorithms with an approximation ratio bounded by a constant. Proof. For the proof, it suffices to note that the P special case of the problem 1|N R, dj = d| Tj P with the optimal value Tj = 0 is N P -hard in the strong sense.

2

Problem 1|N R|

P

Tj

P

For 1|N R, pj = p| Tj , there exists an optimal schedule which has the structure π = (π1 , π2 , . . . , πy ), where the jobs in the partial schedule πi , i = 1, 2, . . . , y, are processed in EDD order. For the special case of problem 1|N R, pj = P p| Tj with g1 ≤ g2 ≤ . . . ≤ gn , d1 ≤ d2 ≤ . . . ≤ dn , schedule π ∗ = (1, 2, . . . , n) is optimal. Next, we consider a more specific situation, namely a sub-problem denoted as 1|N R : αt = P 1, pj = p| Tj (see below). After proving N P hardness of this special case, we consider another special case denoted as 1|N R, G(t) = M, pj = P p| Tj and derive a relation between these two sub-problems. Now we consider the situation, where the times of earnings of the resource are given by P {t1 , t2 , . . . , ty } = {1, 2, . . . , gj }, t1 = 1, t2 = P 2, . . . , ty = gj , and G(ti ) = 1 for i = 1, 2, . . . , y. This condition is denoted as αt = 1 [3]. Therefore, we can denote this problem as 1|N R : αt = 1, pj = P p| Tj .

g

1, 2, . . . , n, where M = n j such that M ∈ Z+ . We denote this special case by 1|N R, G(t) = P M, pj = p| Tj . Two instances of problems 1|N R : αt = P 1, pj = p| Tj and 1|N R, G(t) = M, pj = P p| Tj are called corresponding, if all parameters dj , pj , gj , j = 1, 2, . . . , n, for the two instances are the same.

Lemma 1 There exist two corresponding inTheorem 3 The problem 1|N R : αt = 1, pj = stances of the problems 1|N R : αt = 1, pj = P P P p| Tj is N P -hard. p| Tj and 1|N R, G(t) = M, pj = p| Tj which have different optimal schedules. Proof. We give the following reduction from the P problem 1|| Tj . Given an instance of the prob- Proof. We consider an instance with n = 2 jobs P lem 1|| Tj with processing times p0j and due and p1 = p2 = 1, g1 = 1, g2 = 5, d1 = 7, d2 = P dates d0j for j = 1, 2, . . . , n, we construct an in- 6. For problem 1|N R : αt = 1, pj = p| Tj , we P P P stance of problem 1|N R : αt = 1, pj = p| Tj have Tj (π 1 ) = 0 and Tj (π 2 ) = 1, where π 1 = as follows. Let gj = p0j , pj = 0 and dj = d0j for (2, 1) and π 2 = (1, 2). On the other hand, for the 2

P

problem 1|N R, G(t) = M, pj = p| Tj , we have P P Tj (π 1 ) = 2 and Tj (π 2 ) = 1. Thus, the above two instances have different optimal schedules.

         Now, let dj = 0 for j = 1, 2, . . . , n. For      two corresponding instances of problems 1|N R :    P  αt = 1, pj = 1| Cj and 1|N R, G(t) = M, pj =    P   1| Cj , let Cj (π) be the completion time of job j    

p2n+1 = 1, p2i = M n−i+1 , p2i−1 = p2i + bi , d=

n P

i=1

p2i +

1 2

i = 1, 2, . . . , n, i = 1, 2, . . . , n,

P

bj ,

g = 1, t0 = 0, t1 = d, P t2 = t1 + bj + 1, t3 = t2 + p2n + bn , ... ti = ti−1 + p2(n−i+3) + bn−i+3 , ... tn+1 = tn + p4 + b2 ,

G(t0 ) = n, G(t1 ) = 1,      G(t 2 ) = 1,     G(t  3 ) = 1,  For two corresponding instances of problems    P ...  1|N R : αt = 1, pj = 1| Cj and 1|N R, G(t) =    P G(t  i ) = 1,   M, pj = 1| Cj , we have   ...    G(tn+1 ) = 1. (3) n P Cj0 (π) It is obvious that there are at least n + 1 tardy j=1 jobs in any feasible schedule. We define a canoni< 2. n P Cj (π) cal schedule as a schedule of the form

according to the job sequence π for the first problem and Cj0 (π) be the completion time of the same job according to π for the second problem.

j=1

(V1,1 , V2,1 , . . . , Vi,1 , . . . , Vn,1 , 2n + 1, Vn,2 , . . . , Vi,2 , . . . , V2,2 , V1,2 ), There exists an instance of problems 1|N R : P αt = 1, pj = 1| Cj and 1|N R, G(t) = M, pj = where {Vi,1 , Vi,2 } = {2i − 1, 2i}, i = 1, 2, . . . , n. P Moreover, let π = (E, F ) and for the two par1| Cj for which we have tial schedules E and F , we have |{E}| = n and |{F }| = n + 1. Note that in any canonical schedn P ule, all jobs in sub-sequence F are tardy, the last Cj0 (π) job in sub-sequence E can be tardy or on-time 1 j=1 ≈2− . n P while all other jobs in sub-sequence E are on-time. n C (π) j

j=1

Theorem 5 For instance (3), there exists an optimal schedule which is canonical. We note that in a canonical schedule, there are either n + 1 or n + 2 tardy jobs (job Vn,1 can be tardy or on-time). Moreover, as we prove in the following theorem, in an optimal canonical schedule, there are only n + 1 tardy jobs and thus, all jobs in sub-sequence E are on-time.

P

Theorem 4 The special case 1|N R, pj = p| Tj , where the number of times of earnings of the resource given by t0 , t1 , . . . , ty is less than or equal to n, is N P -hard.

Theorem 6 The instance of the partition problem has an answer ”YES” if and only if in an We give the following reduction from the par- optimal canonical schedule, the equality Special Case 1|N R, dj = d, gj = g|

tition problem. Denote M = (n

n P

P

Tj

bj )n . Let us

X

j=1

consider the following instance with the set of jobs holds. N = {1, 2, . . . , 2n + 1}: 3

j∈E

pj = d

g|

References

Thus, the special case 1|N R, dj = d, gj = Tj is N P -hard.

P

3

[1] J. Carlier and A.H.G. Rinnooy Kan, Scheduling subject to Nonrenewable-Resource Constraints. Oper. Res. Lett. 1(2), 52 – 55, 1982.

Budget Scheduling Problems with Makespan Minimization

[2] R. Slowinski, Preemptive Scheduling of Independent Jobs on Parallel Machines subject to Financial Constraints. European J. Oper. Res. A budget scheduling problem is a financial schedul15, 366 – 373, 1984. ing problem described in this paper, where instead of the values gj , values gj− ≥ 0 and gj+ ≥ 0 are [3] A. Toker, S. Kondakci and N. Erkip, Schedulgiven. The value gj− has the same meaning as gj ing Under a Non-renewable Resource Conin the financial scheduling problem. However, at straint. J. Oper. Res. Soc. 42(9), 811 – 814, the completion time of job j, one has additional 1991. earnings gj+ of the resource. If we have gj− ≥ gj+ for all j = 1, 2, . . . , n, then [4] A. Janiak, One-Machine Scheduling Problems with Resource Constraints. System Modelling the new instance with gj = gj− − gj+ is not equivn and Optimization. Springer Berlin / HeidelP alent to the original one. Let G = (gj− − gj+ ). berg, 358 – 364, 1986. j=1 G(t) < G + max gj− , then not all sequences [5] ∀t (schedules) π are feasible. We denote this problem as 1|N R, gj− , gj+ |Cmax . It is obvious that this problem is N P -hard in the strong sense (since the financial scheduling problem is a special case of the budget scheduling prob[6] lem). If there exists a feasible schedule for an instance of problem 1|N R, gj− , gj+ , gj− > gj+ |Cmax , then the schedule π = (1, 2, . . . , n) with g1+ ≥ g2+ ≥ . . . ≥ [7] gn+ is feasible as well.

If

P

gj−

gj+

A. Janiak, C.N. Potts and T. Tautenhahn, Single Maschine Scheduling with Nonlinear Resource Dependencies of Release Times. Abstract 14th Workshop on Discrete Optimization, Holzhau/Germany, May 2000. J.K. Lenstra, A.H.G. Rinnooy Kan and P. Brucker, Complexity of Machine Scheduling Problems. Annals Discrete Math. 1, 343 – 362, 1977. M.R. Garey and D.S. Johnson, Computers and Intractability: The Guide to the Theory of N P -Completeness. Freeman, San Francisco, 1979.

If inequality > does not hold for all j = 1, 2, . . . , n, then we can use the following list scheduling algorithm for constructing a feasible schedule. [8] E.L. Lawler, A Pseudopolynomial Algorithm Algorithm A. First, all jobs j ∈ N with gj+ − for Sequencing Jobs to Minimize Total Tardigj− ≥ 0 are scheduled. In particular, schedule ness. Ann. Discrete Math. 1, 331 – 342, 1977. among these jobs the job with the earliest possible starting time, if there is more than one job [9] C.N. Potts and L.N. Van Wassenhove, A Decomposition Algorithm for the Single Machine with this property, select the job with the largest + − + − Total Tardiness Problem. Oper. Res. Lett. 1, value gj − gj . If all jobs j with gj − gj ≥ 0 363 – 377, 1982. have been sequenced, schedule the remaining jobs + according to non-increasing values gi . Lemma 2 The problem 1|N R, gj− , gj+ |Cmax is in AP X. 4